The range of $a\in R$ for which the function $f\left(x\right)=(4a-3)(x+{\log }_{e}5)+2(a-7)\cot \left(\frac{x}{2}\right){\sin }^2\left(\frac{x}{2}\right), x\ne 2n\pi ,n\in N$ has critical points, is:

$f\text{'}\left(x\right)=4a=3+(a-7)\cos x=0$

Should have solution

So $\frac{3-4a}{7-a}\in [-1,1]$

$\frac{3-4a}{7-a}\ge -1$

$\frac{3-4a+7-a}{7-a}\ge 0$

$\frac{2-a}{7-a}\ge 0$

$\therefore a\in (-\infty ,2]\cup [7,\infty )$

$\frac{3-4a}{7-a}\le 1$

$\frac{3-4a-7+a}{7-a}\le 0$

$\frac{3a+4}{a-7}\le 0, a\in [-\frac{4}{3},7]$

Take intersection $\therefore a\in [-\frac{4}{3},2]$